Decorated vertices with 3-edged cells in 2D foams: exact solutions and properties.

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solutions and properties.

Marco Mancini, Christophe Oguey

To cite this version:

Marco Mancini, Christophe Oguey. Decorated vertices with 3-edged cells in 2D foams: exact solutions and properties.. 2007. �hal-00108005v3�

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hal-00108005, version 3 - 6 Mar 2007

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Decorated vertices with 3-edged cells in 2D foams: exact solutions and properties.

M. Mancini^1aand C. Oguey²

1 GMCM, CNRS UMR 6626, Universit´e de Rennes I, 35042 Rennes cedex, France.

2 LPTM, CNRS UMR 8089, Universit´e de Cergy-Pontoise, 95031 Cergy-Pontoise, France.

v5.3 March 6, 2007/ Received: date/ Revised version: date

Abstract. The energy, area and excess energy of a decorated vertex in a 2D foam are calculated. The general shape of the vertex and its decoration are described analytically by a reference pattern mapped by a parametric Moebius transformation. A single parameter of control allows to describe, in a common framework, different types of decorations, by liquid triangles or 3-sided bubbles, and other non-conventional cells. A solution is proposed to explain the stability threshold in the flower problem.

PACS. 83.80.Iz Emulsions and foams – 82.70.Rr Aerosols and foams – 82.70.Kj Emulsions and suspensions – 68.03.Hj Gas-liquid and vacuum-liquid interfaces: Structure, measurements and simulations

1 Introduction

Since J. Plateau [1], 2D foams have been extensively studied, experimentally and theoretically, because they are simpler than three-dimensional systems [2].

In the dry model of 2D soap foams, the gas is assumed incompressible and the liquid fraction is assumed vanishing. Although the dry model correctly describes some as- pects of foam physics like energy minima, equilibrium configurations,etc[2], in many other cases, the presence of the liquid needs to be taken into account, both theoretically [3] and to match experimental observations [4,5,6].

The more realistic model of foams, including a small liquid fraction at the vertices, is related to the dry model by Weaire’s decoration theorem. The decoration theorem [7] state that, in 2D foams, the films connecting to a Plateau border with vanishing disjoining pressure, if con- tinued inside it, all intersect at a single point and satisfy the equilibrium conditions (fig. 1-a). Conversely, it is always possible to decorate a 3-fold vertex at equilibrium by a Plateau border, that is, to replace the vertex by a (small) triangle of liquid without changing the geometry outside the triangle and still satisfying equilibrium.

The star-triangle equivalence [3] extends the decoration theorem to general cellular systems where the surface tensions have arbitrary values; this includes 3-sided bubble in standard foams (constant uniform surface tension) or in non standard foams (where the surface tension takes different values on different films [8,9]), or Plateau borders with non vanishing disjoining pressure. In this con- text, bubbles, drops and Plateau borders are treated on a common footing ascells.

a e-mail:[email protected]

The decoration theorem validates the ideal dry model for slightly wet foams as far as equilibrium is concerned [2];

the star-triangle equivalence permits, in principle, to take away 3-sided bubbles in searching equilibrium configurations. However, the foam energy, the area of the bubbles, coarsening and most mechanical properties of the foam change if the decoration is switched on or off.

In dry foams, since the energy is the product of surface tension by film length, minimisation of line-length at fixed bubbles area completely determines the 2D foams equilibrium structure. The energy of a progressively strained dry foam increases until two threefold vertices approach one another and undergo a T1 neighbour exchange [10,11,12].

The T1 quickly reduces the energy by a finite amount:

the energy difference between the configuration with an unstable fourfold vertex and the one with a pair of threefold vertices linked by a film. That is why the location and statistics of T1 events determine the inelastic properties of the foam and how it releases energy under strain or quasi-static flow.

If the foam is not dry, the presence of Plateau borders reduces the film length, and thus favours the switch compared to a dry foam [13]. At the switch, two triangular Plateau borders meet, merge [14,15] and then split to a new configuration involving again two distinct Plateau borders. So, in terms of energy furnished to the foam, the presence of Plateau borders reduces the barrier one needs to overcome to trigger a T1. Furthermore, it is well known that the yield stress of a foam decreases with the liquid fraction [16,17,18,19].

By similar considerations, when 3-sided bubbles are inserted at some vertices of the foam, one can expect an

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increase of the T1 energy barrier and, therefore, a reduction of the shear plasticity due to edge flips.

In this paper, we calculate the energy of a wet-decorated foam starting from the energy of the dry model.

Recently, Teixeira and Fortes [20] gave the equations describing the exact geometry of a general Plateau border with zero disjoining pressure: the surface tension of the films and of the liquid-gas interfaces are related by γfilm= 2γborder; they calculated the excess energy among other quantities. The excess energy is the energy difference between a decorated and a dry vertex.

Using the invariance of 2D foams under Moebius transformations [21], we generalise this problem to the cases where the surface tension of the films and that of the decorating triangles are arbitrary, giving the Plateau border and 3-sided bubble problems a unified description. We calculate the excess energyEand all the geometrical quantities as functions of four parameters which characterise the cluster size and shape. Normalising the excess energy by the square root of the triangle area gives a scale invariant quantityǫβ: therelative excess energy.

In realistic foams, the relative excess energy of 3-cells is found to be approximately linear as a function of the triangle area. The slope ofǫβ depends on the form of the decorated films. According to Lewis’ law [22], the average cell area increases linearly with the number of sides or neighbours:hAni ∝n. So the triangle areaA3 is small on average. As we’ll see, the slope is proportional to the sum of the squared curvatures of the films meeting at the decorated vertex. Being zero in the 3-fold symmetric case, the slope ofǫβmeasures the deviation from perfect symmetry.

In the final part, we apply our analytical results to the flower problem. In the experiments, a bubble spon- taneously gets out [5] before the critical area is reached where the dry model predicts a spontaneous symmetry reduction [23]. Following the experimentalists’ suggestion that the ejection might be due to the presence of liquid, we solve the problem by including liquid triangles around the vertices and calculating a threshold area for the cen- tral bubble at which two vertices get into contact. Our conjecture is that this contact triggers the ejection.

The paper is organised as follows: Section 2 recalls the equilibrium equations for 2D foams with non-constant surface tensions and the star-triangle equivalence. Section 3.1 defines the reference structure which is then used, in sec.

3.2 and 3.3, to describe general 3-cells, including Plateau borders, 3-sided bubbles and similar patterns. Section 4 contains the calculation of the main quantities: the energy gap between bare and decorated structures, the area of the 3-cell and the relative excess energy. In section 4.4, the relative excess energy is studied as a function of the pressure of the four bubbles involved. Section 5 contains the series expansion of the relative excess energy with respect to the 3-cell area when the star films are fixed. Fi- nally, in Section 6, we apply our analytical results to the flower problem.

2 Equilibrium of 2D foams

To describe various types of 2D foams, we consider here a general 2D cellular system with surface tensions verifying the following equilibrium properties.

The cells are separated by interfaces obeying Laplace- Young’s law:

∆P+γk= 0. (1)

∆P = P2−P1 is the pressure difference across the interface, γ the surface tension and k the curvature of the interface [2]. In static conditions and in absence of applied field, the pressure inside each cell is constant so that the interfaces are arcs of circles¹.

At equilibrium, the interfaces meet three by three at vertices in a manner satisfying Plateau’s laws [1,2] :

P3

j=1γjt_j= 0, (2)

P3

j=1γjkj= 0, (3)

wheret_j is the unit vector tangent to the interface at the vertex and where the surface tension, γj (γj >0), differs from interface to interface [3,8,9,24].

Equations (2) and (3) imply that the symmetry group of 2D cellular systems at equilibrium is the group of homo- graphies, or Moebius maps, generated by Euclidean sim- ilarities and inversion [24,25]. Some definitions and basic properties of Moebius maps are recalled in appendix A.

Equations (2) and (3) also imply that the centres of curvature of three circular edges meeting at a vertex are aligned [24].

This common framework describes several situations.

A cell containing liquid is a Plateau border. A cell filled with gas is commonly called a bubble. The sides of a liquid cell are liquid-gas interfaces, with liquid-gas surface ten- sionγℓg. The edges separating (gaseous) bubbles are liquid films, containing a negligible amount of liquid compared to Plateau borders. Films have a specific surface tension γfilm. In standard dry equilibrated 2D foams, the surface tension is constant over the entire foam (γj =γ=γfilm).

In non-standard dry 2D foam, the surface tension may vary from film to film (γj 6=γj^′) [8,9]. Slightly wet equilibrated 2D foams contain bubbles and a small amount of liquid mainly confined in Plateau borders forming concave triangles around the bubble vertices.

The energy of a 2D cellular cluster is the sum, over the set of edgesj, of the edge length,lj, weighted by surface (line in 2D) tension:

E=X

j

γjlj. (4)

2.1 Star–Triangle Equivalence

A triangle is a three-sided equilibrated cell. Each vertex is the end of exactly 3 curved edges; two of them are part

1 Even if we admit surface tension varying from edge to edge, we always assume that it remains constant inside every edge.

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of the triangle boundary but the third one is external, outside the triangle. If the foam is viewed as a graph, this edge connects the triangle to the rest of the foam. Each triangle has three such connecting edges, or bonds, one at each vertex.

Astaris an equilibrated figure formed by a vertex and the three edges meeting at it.

At equilibrium the pressure, or the area, and the surface tensions fix the triangle geometry. The star–triangle equivalence states the following [3]:

– For any triangle, the three external bonds, if extended inside the triangle with the same curvature, all intersect at a common point, that forms a star with the extended edges.

– Conversely, given any star, there is a continuous range of areas extending down to 0, and an open set of tension values, such that, for any prescribed values in these ranges, there is a triangle with one vertex on each branch of the star that forms an equilibrated figure once the portion of the star inside the triangle has been removed. Briefly said, the triangle decorates the star.

a)

j^sta j^tri

v₀

b)

j^sta j^tri

v₀

Fig. 1.Star–triangle equivalence in the case of a Plateau border (a) and a three-sided bubble (b). The triangle (solid lines) can be replaced by the star (dotted) and, conversely, the star can be decorated by the triangle, in a way preserving equilibrium. The connecting films (in grey) around the triangle continue, as dotted arcs of circles, inside the triangle to meet at the equilibrated pointv0.

According to this equivalence, the star and triangle can replace one another without any modification in the rest of the foam, and in a way preserving all the equilibrium conditions. To be precise, both figures have the same contact points with the outside foam, namely the triangle vertices;

in the substitution, the star branches are limited to the segments inside the triangle. The external parts are kept unchanged. The star–triangle equivalence was proved in full generality for cellular systems with multiple surface tensions [3].

When we consider a triangle and its associated star as a whole, astar+triangle, we will call it3-cell. We need the complete figure to evaluate the excess energy, for example.

But it should stay clear that, physically, the star and triangle cannot be both present at the same time. The only exceptions will be found in sec. 3.3.

The star–triangle equivalence contains Weaire’s decoration theorem [7] by triangular Plateau borders as a particular case (fig. 1-a): when the surface tension of the triangle sides is that of a liquid-gas interface,γe≡γtri=γℓg, while the star has the film surface tension, γi ≡ γsta = γfilm≃2γℓg, at zero disjoining pressure.

For a standard dry foam, star-triangle equivalence in- volves a star and a 3-sided bubble with the same tension everywhere:γi=γe=γfilm.

In 3 dimensions, a decoration-bubble theorem holds only for spherical foams [26]. When the films are not nec- essarily spherical, Teixeira and Fortes proposed a modi- fied, approximate, version involving both line and surface tensions [27].

3 Conformal description of 3-cells

In this section the description of the 3-cell (star+triangle) is given. First, we construct a reference 3-cell depending on two parameters. Then, a general 3-cell is obtained by a Moebius transformation.

To fix notations (see fig.1), letϕ^tri, of radius r^tri, be the edges of the triangle (decorating edges), and ϕ^sta, of radius r^sta, the edges of the star (decorated or internal edges). When the star, with vertex v0, is decorated by a triangle, v0 and ϕ^sta are virtual. Conversely, when the star is not decorated, theϕ^triare virtual. We will use the convention of signed angles and radii, in a way such that the arc length is always positive [7].

In this paper, we allow only two different values for γ: the surface tension on the triangle boundary sides is γe ≡γtri, and that of the external and internal films, on the star edges, is γi≡γsta. These tensions are related by equation (2), applied to a vertex of the triangle:

γi= 2γecosα. (5) Thecontact angleαis the angle between ϕ^sta andϕ^tri.

If the thickness of the films, h, is not negligible, the angleαis related to the disjoining pressureΠ [8,9,26]:

Π = γi

h(cosα−1).

The excess energy is the energy gained by decorating the vertex. It is the difference of the internal and triangle edge lengths, weighted by the surface tensions [20]:

E≡Etriangle−Estar=γeL^tri−γiL^sta

=γi

L^tri

2 cosα−L^sta

. (6)

Therelative excess energyis defined as the ratio of the excess energyE over the square root of the triangle area:

ǫ≡ E

A^1/2_triangle. (7)

From now on, we will set γi = 1; the surface tension of the internal films is our tension unit.

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3.1 The reference 3-cell

The reference 3-cellC(q, β), of parametersβ andq, is the completely symmetric star+triangle defined in figure 2.

Α V

C

q Β

j^tri

r^tri= q

2 sinΒ

r^sta=¥

j^sta

v0=0 q 1

!!!3

Α V

C

Β q

j^tri r^tri

j^sta

v₀=0 q

!!!3

Fig. 2. The reference, symmetric, 3-cell with parameters β andq. The star (internal films) is represented by dashed lines.

V Cis a radius of the left side. Solving the triangleV OCshows that the angle atCis indeedβ=α−π/6. The case of a bubble, β >0, is on top; the Plateau border case,β <0, is below.

The parameterq≥0 is the chord length of the curved sides of the reference triangle. It specifies the triangle size.

The parameter β is the angle defined by β ≡α−π/6.

As conformal means angle preserving, the contact angle α, and soβ, are preserved by the Moebius transformation applied to, or from, the reference cell. Choosingβ, rather thanα, helps in treating Plateau borders and 3-sided bubbles in a common way. In figure (1-a),β is equal toπ/6;

in fig. (1-b), β = −π/6. In the reference cell, the angle subtended by the triangle sides equals |2β|; this property is not generally preserved by conformal transformations.

Whenβ=π/3,γi/γe= 0 and the sides of the triangle C(q, β) form a circle. This case describes a 3 cells partition of a disc surrounded by a rigid membrane. So, the interpretation of the triangle as a bubble or a liquid drop is limited to the valuesβ ∈ [−π/6, π/3].

The edge curves are parametrised by ϕ^sta_j (t1) =e^iθ^jt1, witht1∈[0, q/√

3], (8)

ϕ^tri_j (t2) = qe^iθ^j 2 sin(β)

2

√3cos(β+π/6)−e^{i β t}²

, (9)

with t2 ∈ [−1,1];j = 1,2,3 indexes the edges and θj ≡ 2π j/3.

The curve (9) depends continuously onβ∈[−π/6, π/3].

Its curvature vanishes, and changes sign, at β = 0. The equilateral triangle is curved, convex, at positiveβ; curved, concave, at negativeβ; and it has straight edges atβ = 0.

Atβ = 0 the parametrisation is given by the limit of (9) asβ →0:

ϕ^tri_j (t2) =−qe^iθ^j1 2

3^−1/2+i t2

. (10)

With (6), the excess energy of the reference cell is Eβ(q) =

3β

2 sinβcos (β+^π₆)−√ 3

q . (11) The triangle area is the sum of the area, A△, of the rectilinear equilateral triangle based on the same vertices, and A∪, the signed area of the three lenses around the straight triangle:

Aβ(q) =A△+A∪=

√3 4 +3

8

2β−sin(2β) sin²β

!

q². (12) Finally, the reference relative excess energy is obtained by dividing the excess energy by the area square root:

ǫ⁰_β= Eβ(q) Aβ(q)^1/2

= sign(β) q

3β−2√

3 cos (β+^π₆) sinβ

cos (β+^π₆) . (13) Being dimensionless,ǫ⁰_βdoes not depend onq. In the particular cases of a Plateau border and of a 3-sided bubble, the respectiveǫ⁰_β are

ǫ⁰_−π/6=− q√

3−π/2≃ −0.401565, ǫ⁰_π/6=

q

2(π−√

3)≃1.67901.

As a final remark on the reference, notice that the star is fixed, made of straight edges meeting at 2π/3, the standard Plateau angles. In particular, the star is independent of the parameter β and it is affected by q only regarding the length at which its edges are cut. This indicates why, in all subsequent formulae, the quantities concerning only the star do not depend on β, nor on q except for side length.

3.2 The general 3-cell

Applying a suitable Moebius transformation to the reference 3-cell produces a general 3-cell at equilibrium (the star and triangle both satisfy eqs. (1), (2) and (3)).

Given a star, the decorating triangle is entirely speci- fied by its areaAand contact angleα; those will be related

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to the parametersqandβof the reference 3-cell. Now, our transformation must map the (fixed) reference star onto a general star of any possible shape (at equilibrium, of course). But, in the star, centred at vertex ˜v⁰, the three (internal) films i = 1,2,3 have curvature ki. Because of equation (3), only two of the curvatures are independent.

So, removing the Euclidean degrees of freedom, we need only a one-complex parameter Moebius transformation to reach all possible star+triangle figures.

A general Moebius transformationf, given by (44), has six real parameters, much beyond our needs. We require that the transformation i) preserves orientation and ii) maps the interior of the reference 3-cell onto the interior of the general 3-cell. In addition, we can fix a length scale, the star vertex position, and one of the film tangents there (by conformality, this fixes all the tangents at v⁰). Choosing the origin atv⁰, this meansf(0) = 0 andf^′(0)>0. Then the Moebius transformation can be written in the form

fs(z) = (1− |s|²) z

1−¯s z, with|s|²=ss <¯ 1, (14) depending on the single complex parameters. The result is a 4 (real) parameters star+triangle figure:Ce(s, q, β) = fs(C(q, β)). We designate the transformed quantities by a tilde (appendix A). An example is given in figure 3.

a) b)

Fig. 3. a) The reference 3-cell C(q, β), β = π/6 (in black) and β = −π/6 (in grey). b) General 3-cell ˜C(s, q, β), image of a) by fs. The figures are calculated for q = √

2 and s = 0.2 exp (−i^π5). The circumscribed circle in b) is given by (15).

The pointsz= 0 andsare fixed points offs. The complex function fs maps the reference circumscribed circle, of centreC0=v0= 0 and radiusr0=q/√

3 (fig. 3), onto the circle of parameters (eq. (45)):

( ˜C0,˜r0) = q²s(1− |s|²) 3−q²|s|², q

√3(1− |s|²)

|3−q²|s|²|

!

. (15) The 3-cell is contained in the disc ( ˜C0,˜r0) for |s|q <

√3, while it is outside²the disc for|s|q >√ 3.

As the equilibrium equations (2) and (3) are left invariant byfs, the image of the reference, either as a star or as a triangle, is in mechanical equilibrium.

In order to get physically meaningful patterns,qmust be bounded: q < qmax(s, β). An expression forqmax(s, β)

2 This peculiar situation is possible only when the triangle is curved and has two concave sides.

will be derived in section 4.1. Here, we just give its origin and meaning. At q = qmax, the reference boundary,

∂C(q, β), meets 1/s, the pole of¯ fs; this condition defines qmax(s, β). When q < qmax, the reference triangle is mapped to a bounded triangle but, whenq > qmax, the reference interior is mapped to the exterior of the triangle boundary, of infinite area, violating our requirement ii), above.

The divergence at q → qmax is illustrated in figure 4. First, recall that any equilibrated star vertex has a

q q’

v~ 0

q=qmax

v~ 0

*

Fig. 4. Two different 3-cells ˜C(s, q, β) calculated forβ=π/6, q and q^′, withq^′ > q. The continuations of the internal films is plotted in grey (green on-line).

conjugate v^∗ where the circles supporting the star edges also meet. The position of v^∗ is the mirror reflection ofv through the line of the curvature centres [24,26]. For the reference C(q, β), the conjugate ofv0 = 0 is v^∗₀ = ∞; in Ce(s, q, β), ˜v^∗₀ is given by

˜

v^∗₀=fs(∞) =−s(|s|⁻²−1). (16) By increasing q alone, the triangle vertices move along the star edges, towards infinity in the reference, and so, towards ˜v^∗₀ under the mapping fs (figure 4). When q ≥ qmax, the triangle area is infinite and all the edge radii are negative.

In real foams, where the star edges end at neighbour vertices, this divergence is not reached. The route to it is deviated by topological changes. Indeed, suppose that the area of a 3-sided bubble is increased progressively, for example by injecting gas; much before one sees its area di- verging, topological changes will occur, either transform- ing the triangle into a higher polygon, or breaking films.

3.3 Extension to special 3-cells

Although the 3-cell is well defined only for values−π/6≤ β ≤ π/3, the equations can be extended, for particular values of the parameters, to cover other physical or math- ematical situations. These cases are special because the star and triangle are taken here altogether to form a complete equilibrated cluster, without any outside foam. With contracting tensions, this is only possible for obtuse contact angles, that is, β≥π/3.

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3.3.1 Caseβ=π/3: triple partition of the disc

As noticed in sec. 3.1, the referenceC(q, β =π/3) is a disc.

fs maps it to a disc divided into three bubbles by films meeting the outer boundary orthogonally³. Ca˜nete and Ritor´e [28] proved that this type of graph is the unique least-perimeter way of partitioning the unit disk into three regions of prescribed areas.

Fig. 5.Forq=√

3 andβ=π/3, varyings, the star describes all the least-perimeter partitions of the unit disk into three regions of prescribed areas.

3.3.2 Caseβ=π/2: three bubble cluster

Whenβ =π/2, consideringϕ^staandϕ^trias real films, the graphics of the reference 3-cell is equivalent to a symmetric three bubble cluster (figure 6-a). The absolute value of (13) gives its relative energy (E/√

A):

ǫ3=|ǫ⁰_π/2|= (4√

3 + 6π)^1/2≃5.07718.

Applying (14) to the reference produces a general three bubble cluster with constant surface tension (figure 6-b).

a)

Α Β

0 r

!!!3

b)

0

Fig. 6. a) At β=π/2, the reference describes a cluster of 3 identical bubbles. b) Three-bubble cluster transformed byfs.

4 Area and energy of a general 3-cell

To calculate the relative excess energy of a 3-cellCe(s, q, β), we need to calculate the star and triangle perimeters, its area and the edge curvatures.

3 α = π/2 is the angle formed by the soap films with a smooth rigid wall or membrane.

Let us start with the edge radii. Using equations (45) and (46), they are, for j= 1,2,3,

˜

r^sta_j = 1− |s|²

2|s|sin(θj−θs), (17)

˜

r_j^tri= 6 1− |s|² qsinβ

4|3¹²sinβ−q¯se^iθ^jcos(β+^π₆)|²−3q²|s|², (18) whereθs= arg(s). As anticipated (sec. 3.1), the star radii

˜

r^sta = ˜r^sta(s) don’t depend on q. The radii inverse, that is, the curvatures, verify the equilibrium conditions (3), and the following equations:

K_sta² (|s|)≡ X3 j=1

1 (˜r^sta_j )² = 6

|s| 1− |s|²

2

, (19)

K_tri² (|s|, β)≡ X3 j=1

1

(˜r^tri_j )² = 8|s|²cos²(^π₆ +β) (1− |s|²)² + + 4 |s|²q²cos(^π₆ −β)−3 sinβ2

3(1− |s|²)²q² . (20) Thus, the sum of the squared curvatures, in the star or in the triangle, doesn’t depend on the argument ofs. The only relevant parameter left for the star sum in eq. (19) is

|s|.

4.1 Perimeters

The value of the internal perimeter ˜L^sta(s, q) follows from a straightforward calculation detailed in appendix B.

Le^sta(s, q) = X3 j=1

˜l^sta_j = X3 j=1

˜

r^sta_j (s) ˜ω^sta_j (s, q), (21)

where

˜

ω_j^sta(s, q) = 2 arctan

3^−1/2q|s| −cos(θj−θs) sin(θj−θs)

+ + 2 arctan

cos(θj−θs) sin(θj−θs)

(22) is the (signed) angle subtended by the internal film ˜ϕ^sta_j (fig. 7). Once more, the internal perimeter Le^sta doesn’t depend onβ.

Analogously, the triangle perimeter is given by:

Le^tri_β (s, q) = X3 j=1

˜l^tri_j = X3 j=1

˜

r_j^tri(s, q, β) ˜ω^tri_j (s, q, β) (23) where ˜ω^tri_j (s, q, β) is the angle subtended by the triangle edge ˜ϕ^tri_j , defined by (55). The angles ˜ω^tri_j verify the equa-

tion X3

j=1

˜

ω_j^tri= 6β , (24)

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2Α ÈΩ^~^triÈ2

ÈΩ^~^staÈ

Fig. 7.Star and triangle forβ=−π/10. Summing the internal angles of the (solid) grey triangle gives (24).

consequence of the fact that the angles of the (rectilinear) triangle sum up to π(see figure 7).

The limit valueqmax(s, β) (sec. 3.2), where the boundary of the reference triangle meets the pole 1/¯soffs, can be derived from (18). Indeed, fixing all the parameters (s andβ) butq, the sign ofkj= 1/˜r^tri_j changes when the circle supportingϕ^tri_j meets the pole 1/¯s. For eachj = 1,2,3, this occurs at the positive solutionq=qj of

kj= 1/˜r^tri_j (s, q, β) = 0. (25) The explicit expression ofqj is:

qj(s, β) =

√3

3−4 cos²(β+^π₆) sin²(θj−θs)1/2

2|s|cos(β−^π₆) −

−

√3 cos(β+^π₆) cos(θj−θs)

|s|cos(β−^π₆) . (26) When q increases, the pole is first met by circle por- tions outside the triangle. The bound q=qmax(s, β) corresponds to the encounter with the triangle boundary in the last of the three circles:

qmax(s, β) = max

j=1,2,3qj. (27)

For q > qmax, the interior of the 3-cell is no more bounded and relation (24) is no longer verified, as illustrated on fig. 4.

4.2 Area

The area of the 3-cell Ce(s, q, β) is the sum of four parts:

the rectilinear triangle based on the triangle vertices and the (signed) area of the three lenses around:

Aeβ(s, q) =Ae△(s, q) +Ae∪(s, q, β) (28) with

Ae△= 2 ˜r^tri₁ r˜^tri₂ sin ^ω^˜₂¹

sin ^˜^ω₂² sin ^π₃ −β+^ω^˜₂³, (29) Ae∪= ¹₂P3

j=1(˜r^tri_j )²(˜ωj−sin ˜ωj). (30) In these equations, ˜ωj stands for ˜ω_j^tri. Althoughβappears in the RHS of (29), it is clear that the area of the triangle Ae△ doesn’t depend on it. Indeed, the triangle△ is built only on the vertices, independent ofα, and soβ(they are determined byqin the reference and then mapped byfs).

4.3 Relative excess energy

Using (21), (23) and (28), we obtain the relative excess energy for a general 3-cellCe(s, q, β):

ǫβ(s, q) =Le^tri_β (s, q)/(2 cos(β+π/6))−Le^sta(s, q) Aeβ(s, q)^1/2 . (31) Before we draw the plots, let us make a change of vari- ables to take advantage of the dilation invariance of the relative excess energy, already noticed. The new parametrisation is defined as follows:





ρ =q(1− |s|²) with ρ≥0,

η =q|s| with 0≤η≤η(β, θ¯ s), θs= arg(s),

(32) where ¯η(β, θs) =|s|qmax(s, β) is independent of|s|by (26).

The fourth parameter,β, is unchanged. The parameter ρ corresponds to a magnification of the 3-cell, so the relative excess energy doesn’t depend on it. Moreover, from the symmetry of the reference 3-cell under the groupD3

generated by mirrors at π/3, the relative excess energy verifies:

ǫβ(η, θs) =ǫβ(η,2π/3±θs). (33)

Fig. 8. Plot of the relative excess energy of a 3-sided bubble (β = π/6) as a function of η = q|s| for some values of 0 ≤ θs≤π/3. The thick solid line corresponds toθs= 0 where the system has a mirror symmetry. The dot-dashed lines point out the limit values ofǫπ/6 calculated at ¯η.

Figures 8 and 9 show the plots ofǫβ(η, θs) for a 3-sided bubble and a Plateau border, respectively. In both cases, the relative excess energy is minimal for η= 0, i.e. in the symmetrical reference configuration.

In the 3-sided bubble case, for low values of η, ǫπ/6

increases withθswhile, for greaterη, it decreases withθs; the change, (∂/∂θs)ǫ(π/6, η, θs) = 0, occurs at η values close to ¯η(π/6,0). Atη= ¯η(π/6, θs), when the area of the 3-sided bubbles diverges (sec. 3.2), ǫπ/6(¯η(π/6, θs), θs) is constant as a function of θs, meaning that all the curves in fig. 8 end at the same ordinate.

In the Plateau border case (β=−π/6),ǫ−π/6is a non increasing function of θs: (∂/∂θs)ǫ−π/6 ≤ 0. For any θs,

(9)

Fig. 9. The relative excess energy of a Plateau border (β=

−π/6) as a function ofη=q|s|for some values of 0≤θs≤π/3.

The solid line corresponds toθs= 0, where the border has a mirror symmetry.

at η = ¯η(0), the triangle vertices stay on a straight line and the area of the rectilinear triangle is zero: Ae△ = 0.

This implies that atη= ¯η(0), the derivative (∂/∂η)ǫ−π/6

has a singularity, corresponding to the cuspidal points of ǫ−π/6(η, θs) in figure 9. Again, in this case, at η =

¯

η(−π/6, θs) andθs6= 0, when the area of the Plateau border diverges (sec. 3.2), the relative excess energy doesn’t depend onθs:ǫ−π/6(¯η, θs) = 1.772. . .(outside the figure).

4.4 Pressure as coordinates

Solving equations (18) and inserting into the formulae of sec. 4.1-4.3 yields the relative excess energy as a function of the side radii ˜r^tri_j , j = 1,2,3. By Laplace’s equation (1), the curvatures are proportional to the pressure differ- ences. IfP0 denotes the pressure in the 3-cell andPj the pressure in the neighbours j = 1,2,3, we can analyse ǫβ

as a function of the pressures.

By the dilation invariance of ǫβ, we can set the absolute value of one of the sides radii to 1 without loss of generality. So, fixingP0= 0 and ˜r^tri₁ = sign(β), (1) implies











P1=− sign(β) 2 cos(β+π/6),

P2=− 1

2 ˜r₂^tricos(β+π/6),

P3=− 1

2 ˜r₃^tricos(β+π/6).

(34)

In figures 10 and 11 (β =π/6 andβ=−π/6, respectively), we have drawn the parametric plots of the relative excess energy as a function ofP3for different values ofP2. In figure 10, the limit ofǫπ/6(P2, P3) whenP3→ −∞

corresponds to configurations of a non-symmetrical cell when the area of the 3-sided bubble goes to zero. P2 = P3 → −∞ gives the configuration ”rat”, whereas in all the other cases,P3→ −∞at fixedP2, the bubble goes to the configuration ”cat” (fig. 12).

Fig. 10.Standard bubble case,β=π/6 andP¹=−1. Plot of ǫπ/6 versus the pressureP³ for different values ofP².

Fig. 11. Plateau border,β=−π/6 andP1=−1/2.ǫ_−π/6 is plotted with respect toP3 for different values ofP2.

In the Plateau border case (figure 11), when P2 = P3 →+∞, the border goes to a configuration ”fox” (fig.

12), where one of the edges is straight. In the other cases, ǫ−π/6 goes to zero. Obviously, in all cases, bubble or border, equal pressure (P1 =P2=P3) givesǫβ(P1, P1) =ǫ⁰_β.

5 Decorating a fixed star

In this section we consider an ideal dry 2D foam at equilibrium and we ask how much the energy of the foam varies if we replace one vertex by a Plateau border or a triangular bubble.

This question is important in foam rheology [10,13,29].

If the foam is not dry, the film length is reduced by the presence of the Plateau borders (see figure 13), favour- ing the switch. Preliminary calculations on small samples show that the presence of liquid Plateau borders reduces the T1 energy barrier, whereas the presence of triangular bubbles leads to locally more stable quadrangles, thus increasing an effective T1 barrier. Qualitatively, as compared to a bare foam, the net result is a decrease of the yield strain and stress when the foam is decorated by